Let $H_1\subset H_2\subset H_3$ be three continuosly embedded and dense complex Hilbert spaces, where $H_3$ is the dual space of $H_1$ and $A:H_1\longrightarrow H_2$ and $B:H_1\longrightarrow H_3$ be two operators. Let $u\in H_1$ and $v\in K\subset H_1$ and consider the scalar product $$ (Au, v) = (Bu, v).$$
On my notes there is something not clear for me, i.e. the following sentence. "If we suppose $K$ be dense in $H_1$, then we have $Au = Bu$."
I don't understand why it is true. Could anyone please help?
I am sorry if the question is trivial.
Thank you in advance!